ARMA/ARIMA/SARIMA Models

Stationarity, Differencing, Model Fitting, Diagnostics, Forcasting, and Validation

ARIMA - Economic Indicator

I choose the ARIMA model for forecasting economic indicators because it’s useful for forecasting a series where the data points are independent of the seasonal components, which is the case with the economic indicators such as CPI, GDP, USD index, unemployment rate, and mortgage rate. These indicators typically have underlying trends or cycles that ARIMA can address through differencing, making the data stationary before applying auto-regressive and moving average components to capture the relationships in the data.

Data Processing


Stationarity Check

Initial assessments via ACF and Augmented Dickey-Fuller tests indicated that CPI required differencing due to non-stationarity. After converting to a time series and applying logarithmic transformation, first differencing was insufficient in detrending, but second differencing indicated stationarity.

Augmented Dickey-Fuller Test Results:
Test Statistic: -0.8219849   P-value: 0.959648
The time series is not stationary based on the ADF test.


Model Fitting

After second differencing CPI, ACF shows three lags, while PACF shows four. This suggests ARIMA parameters p = [0,1,2,3], d = [2], q = [0,1,2]. I’ll test these for the lowest AIC, BIC, and AICc, and cross-check with auto.arima to forecasting.

   p d q       AIC       BIC      AICc
1  0 2 0 -7964.774 -7959.951 -7964.769
2  0 2 1 -8187.294 -8177.650 -8187.281
3  0 2 2 -8242.219 -8227.752 -8242.193
4  1 2 0 -8055.798 -8046.154 -8055.785
5  1 2 1 -8241.533 -8227.066 -8241.507
6  1 2 2 -8241.209 -8221.920 -8241.165
7  2 2 0 -8126.439 -8111.973 -8126.413
8  2 2 1 -8241.021 -8221.732 -8240.977
9  2 2 2 -8239.420 -8215.309 -8239.354
10 3 2 0 -8156.962 -8137.673 -8156.918
11 3 2 1 -8239.300 -8215.189 -8239.234
12 3 2 2 -8246.314 -8217.381 -8246.222
Model fitting with minimum AIC:
 3, 2, 2, -8246.31447088773, -8217.381286544, -8246.2222645211

Model fitting with minimum AICc:
 3, 2, 2, -8246.31447088773, -8217.381286544, -8246.2222645211

Model fitting with minimum BIC:
 0, 2, 2, -8242.21881165827, -8227.75221948641, -8242.19255345258
Series: ts 
ARIMA(1,2,1)(2,0,0)[12] 

Coefficients:
         ar1      ma1     sar1     sar2
      0.2957  -0.8775  -0.2088  -0.1736
s.e.  0.0413   0.0217   0.0347   0.0362

sigma^2 = 6.977e-06:  log likelihood = 4149.36
AIC=-8288.72   AICc=-8288.66   BIC=-8264.61


Model Diagnostics

The ARIMA(3,2,2) model exhibits a satisfactory fit, evident from the patternless residuals and lack of autocorrelation, but its coefficients are not all statistically significant. In contrast, the ARIMA(0,2,2) model, while equally displaying white noise residuals and minimal autocorrelation, boasts statistically significant coefficients, lending greater weight to its predictive accuracy. The SARIMA(1,2,1)(2,0,0)[12] also presents a strong fit, confirmed by its residuals and significant p-values, and it outperforms the ARIMA(0,2,2) model in terms of lower AIC, BIC, and AICc values. Nonetheless, the simpler ARIMA(0,2,2) is preferred due to its adequate fit and less complexity. Both models are considered robust, with the choice between them hinging on the trade-off between simplicity and statistical thoroughness.


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ARIMA(3,2,2):

initial  value -5.780467 
iter   2 value -5.891674
iter   3 value -5.907543
iter   4 value -5.912536
iter   5 value -5.915068
iter   6 value -5.916815
iter   7 value -5.917127
iter   8 value -5.917161
iter   9 value -5.917183
iter  10 value -5.917240
iter  11 value -5.917443
iter  12 value -5.918386
iter  13 value -5.918952
iter  14 value -5.919150
iter  15 value -5.919410
iter  16 value -5.919542
iter  17 value -5.919619
iter  18 value -5.919813
iter  19 value -5.920216
iter  20 value -5.921607
iter  21 value -5.923292
iter  22 value -5.924930
iter  23 value -5.926276
iter  24 value -5.927671
iter  25 value -5.927976
iter  26 value -5.928484
iter  27 value -5.928495
iter  28 value -5.928883
iter  29 value -5.929187
iter  30 value -5.929601
iter  31 value -5.930039
iter  32 value -5.930560
iter  33 value -5.930828
iter  34 value -5.930967
iter  35 value -5.931038
iter  36 value -5.931050
iter  37 value -5.931054
iter  37 value -5.931054
final  value -5.931054 
converged
initial  value -5.912816 
iter   2 value -5.912840
iter   3 value -5.913011
iter   4 value -5.913090
iter   5 value -5.913144
iter   6 value -5.913184
iter   7 value -5.913272
iter   8 value -5.913464
iter   9 value -5.913743
iter  10 value -5.914388
iter  11 value -5.914622
iter  12 value -5.915632
iter  13 value -5.915658
iter  14 value -5.915698
iter  15 value -5.915748
iter  16 value -5.915912
iter  17 value -5.916262
iter  18 value -5.916392
iter  19 value -5.916708
iter  20 value -5.916835
iter  21 value -5.916916
iter  22 value -5.916925
iter  23 value -5.916927
iter  24 value -5.916927
iter  25 value -5.916930
iter  26 value -5.916931
iter  27 value -5.916931
iter  27 value -5.916931
iter  27 value -5.916931
final  value -5.916931 
converged
$fit

Call:
arima(x = xdata, order = c(p, d, q), seasonal = list(order = c(P, D, Q), period = S), 
    include.mean = !no.constant, transform.pars = trans, fixed = fixed, optim.control = list(trace = trc, 
        REPORT = 1, reltol = tol))

Coefficients:
         ar1      ar2     ar3      ma1     ma2
      1.1965  -0.3280  0.0728  -1.7608  0.7647
s.e.  0.0650   0.0541  0.0367   0.0567  0.0545

sigma^2 estimated as 7.243e-06:  log likelihood = 4129.16,  aic = -8246.31

$degrees_of_freedom
[1] 913

$ttable
    Estimate     SE  t.value p.value
ar1   1.1965 0.0650  18.4168  0.0000
ar2  -0.3280 0.0541  -6.0621  0.0000
ar3   0.0728 0.0367   1.9823  0.0477
ma1  -1.7608 0.0567 -31.0300  0.0000
ma2   0.7647 0.0545  14.0428  0.0000

$AIC
[1] -8.982913

$AICc
[1] -8.982842

$BIC
[1] -8.951396

**************************************************************************
ARIMA(0,2,2):

initial  value -5.758140 
iter   2 value -5.863065
iter   3 value -5.909828
iter   4 value -5.910974
iter   5 value -5.911802
iter   6 value -5.911847
iter   7 value -5.911847
iter   7 value -5.911847
iter   7 value -5.911847
final  value -5.911847 
converged
initial  value -5.911431 
iter   2 value -5.911431
iter   3 value -5.911432
iter   3 value -5.911432
iter   3 value -5.911432
final  value -5.911432 
converged
$fit

Call:
arima(x = xdata, order = c(p, d, q), seasonal = list(order = c(P, D, Q), period = S), 
    include.mean = !no.constant, transform.pars = trans, fixed = fixed, optim.control = list(trace = trc, 
        REPORT = 1, reltol = tol))

Coefficients:
          ma1      ma2
      -0.5605  -0.2529
s.e.   0.0317   0.0330

sigma^2 estimated as 7.327e-06:  log likelihood = 4124.11,  aic = -8242.22

$degrees_of_freedom
[1] 916

$ttable
    Estimate     SE  t.value p.value
ma1  -0.5605 0.0317 -17.7039       0
ma2  -0.2529 0.0330  -7.6707       0

$AIC
[1] -8.978452

$AICc
[1] -8.978438

$BIC
[1] -8.962693

**************************************************************************
auto.arima (1,2,1)(2,0,0)[12]:

initial  value -5.855766 
iter   2 value -5.931096
iter   3 value -5.949949
iter   4 value -5.961356
iter   5 value -5.990703
iter   6 value -6.008407
iter   7 value -6.017910
iter   8 value -6.022383
iter   9 value -6.022796
iter  10 value -6.023102
iter  11 value -6.023114
iter  12 value -6.023116
iter  12 value -6.023116
iter  12 value -6.023116
final  value -6.023116 
converged
initial  value -5.937579 
iter   2 value -5.938757
iter   3 value -5.938837
iter   4 value -5.938930
iter   5 value -5.938939
iter   6 value -5.938940
iter   6 value -5.938940
iter   6 value -5.938940
final  value -5.938940 
converged
$fit

Call:
arima(x = xdata, order = c(p, d, q), seasonal = list(order = c(P, D, Q), period = S), 
    include.mean = !no.constant, transform.pars = trans, fixed = fixed, optim.control = list(trace = trc, 
        REPORT = 1, reltol = tol))

Coefficients:
         ar1      ma1     sar1     sar2
      0.2957  -0.8775  -0.2088  -0.1736
s.e.  0.0413   0.0217   0.0347   0.0362

sigma^2 estimated as 6.926e-06:  log likelihood = 4149.36,  aic = -8288.72

$degrees_of_freedom
[1] 914

$ttable
     Estimate     SE  t.value p.value
ar1    0.2957 0.0413   7.1618       0
ma1   -0.8775 0.0217 -40.4427       0
sar1  -0.2088 0.0347  -6.0251       0
sar2  -0.1736 0.0362  -4.7917       0

$AIC
[1] -9.029109

$AICc
[1] -9.029062

$BIC
[1] -9.002845

**************************************************************************
Series: ts 
ARIMA(0,2,2) 

Coefficients:
          ma1      ma2
      -0.5605  -0.2529
s.e.   0.0317   0.0330

sigma^2 = 7.364e-06:  log likelihood = 4124.11
AIC=-8242.22   AICc=-8242.19   BIC=-8227.75

Training set error measures:
                        ME      RMSE         MAE           MPE       MAPE
Training set -2.196212e-05 0.0027077 0.001861392 -0.0006268637 0.04498296
                  MASE         ACF1
Training set 0.0528366 0.0009203494


Equation for ARIMA(0,2,2)v:

\[(1 - B)^2 X_t = (1 + \theta_1 B + \theta_2 B^2) W_t\]


Forecasting

The graph depicts the predicted logarithm of CPI over time, extending from historical data into future projections. The black line represents the actual historical log(CPI) values, showing a general upward trend over time, which indicates that the CPI has been increasing. The blue shaded area starting around 2020 represents the forecasted values, with the shade indicating the confidence interval of the predictions.


Benchmark Method

The ARIMA model forecasts (red line) are closest to the actual data, indicating a superior fit among the methods. Accuracy metrics support this, with ARIMA showing the lowest error rates across the board, suggesting high precision and minimal bias in forecasting. Other models like the Mean, Naive, and Seasonal Naive exhibit higher errors, indicating less accurate predictions. The Drift model performs better than these but is still outclassed by ARIMA. Overall, ARIMA is identified as the best model for forecasting CPI in this case.

ARIMA Model Accuracy Metrics:
                        ME      RMSE         MAE           MPE       MAPE
Training set -2.196212e-05 0.0027077 0.001861392 -0.0006268637 0.04498296
                  MASE         ACF1
Training set 0.0528366 0.0009203494

Mean Model Accuracy Metrics:
                        ME      RMSE       MAE       MPE     MAPE     MASE
Training set -9.418417e-18 0.8627205 0.7913189 -4.129501 19.08201 22.46201
                  ACF1
Training set 0.9974046

Naive Model Accuracy Metrics:
                      ME        RMSE         MAE        MPE       MAPE     MASE
Training set 0.002891558 0.004483797 0.003403071 0.06783973 0.08035247 0.096598
                  ACF1
Training set 0.5749759

Seasonal Naive Model Accuracy Metrics:
                     ME       RMSE        MAE       MPE      MAPE MASE
Training set 0.03439707 0.04412258 0.03522921 0.7988339 0.8224095    1
                  ACF1
Training set 0.9848684

Random Walk with Drift Model Accuracy Metrics:
                       ME        RMSE         MAE          MPE       MAPE
Training set 1.391738e-16 0.003426854 0.002408751 0.0001517385 0.05821684
                  MASE      ACF1
Training set 0.0683737 0.5749759

Model with the best Accuracy Metrics:
 ARIMA ARIMA ARIMA Mean ARIMA ARIMA ARIMA

Data Processing


Stationarity Check

Initial assessments via ACF and Augmented Dickey-Fuller tests indicated that GDP required differencing due to non-stationarity. After converting to a time series and applying logarithmic transformation, first differencing was insufficient in detrending, but second differencing indicated stationarity.

Augmented Dickey-Fuller Test Results:
Test Statistic: -1.049504   P-value: 0.929467
The time series is not stationary based on the ADF test.


Model Fitting

After second differencing GDP, ACF shows one significant lag, while PACF shows five. This suggests ARIMA parameters p = [1,2,3,4,5], d = [1,2], q = [1]. I’ll test these for the lowest AIC, BIC, and AICc, and cross-check with auto.arima to forecasting.

   p d q       AIC       BIC      AICc
1  1 1 1 -1871.924 -1857.043 -1871.791
2  1 2 1 -1861.148 -1849.997 -1861.068
3  2 1 1 -1871.483 -1852.881 -1871.282
4  2 2 1 -1861.697 -1846.829 -1861.563
5  3 1 1 -1871.070 -1848.748 -1870.788
6  3 2 1 -1860.092 -1841.507 -1859.891
7  4 1 1 -1869.752 -1843.710 -1869.375
8  4 2 1 -1859.721 -1837.419 -1859.438
9  5 1 1 -1867.887 -1838.124 -1867.400
10 5 2 1 -1858.660 -1832.640 -1858.281
Model fitting with minimum AIC:
 1, 1, 1, -1871.92391061098, -1857.04266350455, -1871.79057727765

Model fitting with minimum AICc:
 1, 1, 1, -1871.92391061098, -1857.04266350455, -1871.79057727765

Model fitting with minimum BIC:
 1, 1, 1, -1871.92391061098, -1857.04266350455, -1871.79057727765
Series: ts 
ARIMA(0,2,3)(1,0,1)[4] 

Coefficients:
          ma1      ma2      ma3     sar1    sma1
      -0.8748  -0.0010  -0.1093  -0.6082  0.5539
s.e.   0.0572   0.0756   0.0562   0.3597  0.3727

sigma^2 = 0.0001251:  log likelihood = 935.79
AIC=-1859.59   AICc=-1859.31   BIC=-1837.29


Model Diagnostics

The first ARIMA model (1,1,1) shows a good fit with the lowest information criteria scores, indicating effective parameter use. The residuals suggest the model captures the data’s underlying process well. The second model, which appears to be a SARIMA given the seasonal components, is more complex and doesn’t offer a significantly better fit, as the information criteria scores are marginally higher and some coefficients are not statistically significant. The first model is preferable for its simplicity and performance.


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ARIMA(1,1,1):

initial  value -4.489124 
iter   2 value -4.494637
iter   3 value -4.497303
iter   4 value -4.497428
iter   5 value -4.499701
iter   6 value -4.500276
iter   7 value -4.500679
iter   8 value -4.500764
iter   9 value -4.500774
iter  10 value -4.500792
iter  11 value -4.500792
iter  12 value -4.500792
iter  13 value -4.500793
iter  14 value -4.500793
iter  14 value -4.500793
iter  14 value -4.500793
final  value -4.500793 
converged
initial  value -4.500762 
iter   2 value -4.500768
iter   3 value -4.500777
iter   4 value -4.500778
iter   5 value -4.500778
iter   6 value -4.500780
iter   7 value -4.500781
iter   8 value -4.500781
iter   8 value -4.500781
final  value -4.500781 
converged
$fit

Call:
arima(x = xdata, order = c(p, d, q), seasonal = list(order = c(P, D, Q), period = S), 
    xreg = constant, transform.pars = trans, fixed = fixed, optim.control = list(trace = trc, 
        REPORT = 1, reltol = tol))

Coefficients:
         ar1      ma1  constant
      0.4412  -0.3052    0.0076
s.e.  0.2101   0.2191    0.0008

sigma^2 estimated as 0.0001232:  log likelihood = 939.96,  aic = -1871.92

$degrees_of_freedom
[1] 302

$ttable
         Estimate     SE t.value p.value
ar1        0.4412 0.2101  2.1002  0.0365
ma1       -0.3052 0.2191 -1.3930  0.1646
constant   0.0076 0.0008  9.6004  0.0000

$AIC
[1] -6.137455

$AICc
[1] -6.137194

$BIC
[1] -6.088664

**************************************************************************
auto.arima (0,2,3)(1,0,1)[4]:

initial  value -4.209452 
iter   2 value -4.384929
iter   3 value -4.442731
iter   4 value -4.485373
iter   5 value -4.486556
iter   6 value -4.489059
iter   7 value -4.490679
iter   8 value -4.493028
iter   9 value -4.494280
iter  10 value -4.496736
iter  11 value -4.497414
iter  12 value -4.501068
iter  13 value -4.501074
iter  14 value -4.501481
iter  15 value -4.501831
iter  16 value -4.502053
iter  17 value -4.502333
iter  18 value -4.502676
iter  19 value -4.502758
iter  20 value -4.502770
iter  21 value -4.502770
iter  21 value -4.502770
iter  21 value -4.502770
final  value -4.502770 
converged
initial  value -4.497003 
iter   2 value -4.497051
iter   3 value -4.497159
iter   4 value -4.497198
iter   5 value -4.497207
iter   6 value -4.497209
iter   6 value -4.497209
iter   6 value -4.497209
final  value -4.497209 
converged
$fit

Call:
arima(x = xdata, order = c(p, d, q), seasonal = list(order = c(P, D, Q), period = S), 
    include.mean = !no.constant, transform.pars = trans, fixed = fixed, optim.control = list(trace = trc, 
        REPORT = 1, reltol = tol))

Coefficients:
          ma1      ma2      ma3     sar1    sma1
      -0.8748  -0.0010  -0.1093  -0.6082  0.5539
s.e.   0.0572   0.0756   0.0562   0.3597  0.3727

sigma^2 estimated as 0.0001227:  log likelihood = 935.79,  aic = -1859.59

$degrees_of_freedom
[1] 299

$ttable
     Estimate     SE  t.value p.value
ma1   -0.8748 0.0572 -15.2823  0.0000
ma2   -0.0010 0.0756  -0.0137  0.9891
ma3   -0.1093 0.0562  -1.9463  0.0526
sar1  -0.6082 0.3597  -1.6908  0.0919
sma1   0.5539 0.3727   1.4861  0.1383

$AIC
[1] -6.117068

$AICc
[1] -6.116405

$BIC
[1] -6.043705

**************************************************************************
Series: ts 
ARIMA(1,1,1) 

Coefficients:
         ar1      ma1
      0.9997  -0.9836
s.e.  0.0006   0.0103

sigma^2 = 0.0001276:  log likelihood = 934.67
AIC=-1863.34   AICc=-1863.26   BIC=-1852.18

Training set error measures:
                        ME       RMSE         MAE          MPE       MAPE
Training set -0.0001737751 0.01123841 0.007111044 -0.001027974 0.08112324
                  MASE      ACF1
Training set 0.2069521 0.1241003


Equation for ARIMA(1,1,1):

\[(1 - \phi B)(1 - B)X_t = (1 + \theta B)W_t\]


Forecasting

The forecast shows a projected increase in the log-transformed GDP, with historical data indicating a good model fit. The widening confidence intervals suggest greater uncertainty in the longer term. While useful for economic planning, these predictions rely on past trends continuing unchanged and may not account for unforeseen economic events.


Benchmark Method

The ARIMA model seems to closely follow the actual trend, along with the Drift model. The other models—Mean, Naive, and Seasonal Naive—diverge from the actual trend as time progresses, indicating less accuracy.

From the accuracy metrics given, the ARIMA model outperforms the others with the lowest errors across multiple measures (RMSE, MAE, MPE, MAPE, MASE, and ACF1). The Mean Model performs the worst, with the highest errors. The Naive and Seasonal Naive models also show higher errors than ARIMA but are better than the Mean Model. The Random Walk with Drift Model has metrics comparable to the ARIMA model, suggesting it is also a good fit for the data. Overall, the ARIMA and Drift models are indicated as the best for this dataset based on the provided metrics.

ARIMA Model Accuracy Metrics:
                        ME       RMSE         MAE          MPE       MAPE
Training set -0.0001737751 0.01123841 0.007111044 -0.001027974 0.08112324
                  MASE      ACF1
Training set 0.2069521 0.1241003

Mean Model Accuracy Metrics:
                       ME      RMSE       MAE        MPE     MAPE     MASE
Training set 1.560046e-17 0.6856741 0.5943983 -0.6000709 6.726537 17.29873
                  ACF1
Training set 0.9903157

Naive Model Accuracy Metrics:
                      ME       RMSE        MAE        MPE      MAPE      MASE
Training set 0.007608786 0.01356279 0.01029104 0.08636799 0.1165192 0.2994994
                  ACF1
Training set 0.1336013

Seasonal Naive Model Accuracy Metrics:
                     ME       RMSE        MAE       MPE     MAPE MASE     ACF1
Training set 0.03061763 0.04003529 0.03436081 0.3470782 0.388953    1 0.798235

Random Walk with Drift Model Accuracy Metrics:
                       ME       RMSE       MAE         MPE       MAPE      MASE
Training set 3.814826e-16 0.01122745 0.0071468 0.001228953 0.08148863 0.2079927
                  ACF1
Training set 0.1336013

Model with the best Accuracy Metrics:
 ARIMA Drift ARIMA Mean ARIMA ARIMA ARIMA

SARIMA - Treasury Security

SARIMA models are used for treasury yields like 3-month T-bills or 20-year T-bonds because they help predict the regular ups and downs that happen throughout the year. These ups and downs can be due to when the government borrows more, changes in how often people invest, and rules that banks follow at certain times. SARIMA can catch these patterns, making it easier to guess where yields will go next, which is very important for people who invest in these securities.

Data Processing


Stationarity Check

Initial assessments via ACF and Augmented Dickey-Fuller tests indicated that 3Mon T-Bill Yield required differencing due to non-stationarity. After converting to a time series, first differencing and seasonal differencing alone were insufficient in detrending, but the combination of non-seasonal and seaonal differencing indicated strong stationarity.

Augmented Dickey-Fuller Test Results:
Test Statistic: -3.318366   P-value: 0.07045011
The time series is not stationary based on the ADF test.


Model Fitting

After second differencing 3 Month T-Bill Yield, ACF shows three lags, while PACF shows four. This suggests ARIMA parameters p = [0,1,2,3], P = [2], d = [1], [D = 1], q = [0,1,2], Q = [1]. I’ll test these for the lowest AIC, BIC, and AICc, and cross-check with auto.arima to forecasting.

Minimum AIC: 1, 1, 1, 2, 1, 1, -77.5526287298902, -58.9901275250496, -77.0141671914286 
Minimum BIC: 1, 1, 1, 2, 1, 1, -77.5526287298902, -58.9901275250496, -77.0141671914286 
Minimum AICc: 1, 1, 1, 2, 1, 1, -77.5526287298902, -58.9901275250496, -77.0141671914286 
Series: ts 
ARIMA(3,1,0)(2,0,0)[4] 

Coefficients:
          ar1      ar2      ar3     sar1     sar2
      -0.0812  -0.0820  -0.2835  -0.1593  -0.0340
s.e.   0.0814   0.0789   0.0780   0.0905   0.0897

sigma^2 = 0.03345:  log likelihood = 49.13
AIC=-86.25   AICc=-85.73   BIC=-67.54


Model Diagnostics

 [1] "Call:"                                                                                                     
 [2] "arima(x = xdata, order = c(p, d, q), seasonal = list(order = c(P, D, Q), period = S), "                    
 [3] "    include.mean = !no.constant, transform.pars = trans, fixed = fixed, optim.control = list(trace = trc, "
 [4] "        REPORT = 1, reltol = tol))"                                                                        
 [5] ""                                                                                                          
 [6] "Coefficients:"                                                                                             
 [7] "         ar1      ma1     sar1     sar2     sma1"                                                          
 [8] "      0.8182  -1.0000  -0.1960  -0.1040  -0.8524"                                                          
 [9] "s.e.  0.0520   0.0256   0.0993   0.0982   0.0615"                                                          
[10] ""                                                                                                          
[11] "sigma^2 estimated as 0.03124:  log likelihood = 44.78,  aic = -77.55"                                      
[12] ""                                                                                                          
[13] "$degrees_of_freedom"                                                                                       
[14] "[1] 158"                                                                                                   
[15] ""                                                                                                          
[16] "$ttable"                                                                                                   
[17] "     Estimate     SE  t.value p.value"                                                                     
[18] "ar1    0.8182 0.0520  15.7209  0.0000"                                                                     
[19] "ma1   -1.0000 0.0256 -39.1200  0.0000"                                                                     
[20] "sar1  -0.1960 0.0993  -1.9733  0.0502"                                                                     
[21] "sar2  -0.1040 0.0982  -1.0593  0.2911"                                                                     
[22] "sma1  -0.8524 0.0615 -13.8632  0.0000"                                                                     
[23] ""                                                                                                          
[24] "$AIC"                                                                                                      
[25] "[1] -0.475783"                                                                                             
[26] ""                                                                                                          
[27] "$AICc"                                                                                                     
[28] "[1] -0.4734384"                                                                                            
[29] ""                                                                                                          
[30] "$BIC"                                                                                                      
[31] "[1] -0.3619026"                                                                                            

 [1] "Call:"                                                                                         
 [2] "arima(x = xdata, order = c(p, d, q), seasonal = list(order = c(P, D, Q), period = S), "        
 [3] "    xreg = constant, transform.pars = trans, fixed = fixed, optim.control = list(trace = trc, "
 [4] "        REPORT = 1, reltol = tol))"                                                            
 [5] ""                                                                                              
 [6] "Coefficients:"                                                                                 
 [7] "          ar1      ar2      ar3     sar1     sar2  constant"                                   
 [8] "      -0.0871  -0.0869  -0.2877  -0.1652  -0.0364    0.0067"                                   
 [9] "s.e.   0.0816   0.0790   0.0780   0.0908   0.0896    0.0080"                                   
[10] ""                                                                                              
[11] "sigma^2 estimated as 0.03231:  log likelihood = 49.47,  aic = -84.94"                          
[12] ""                                                                                              
[13] "$degrees_of_freedom"                                                                           
[14] "[1] 161"                                                                                       
[15] ""                                                                                              
[16] "$ttable"                                                                                       
[17] "         Estimate     SE t.value p.value"                                                      
[18] "ar1       -0.0871 0.0816 -1.0681  0.2871"                                                      
[19] "ar2       -0.0869 0.0790 -1.0994  0.2733"                                                      
[20] "ar3       -0.2877 0.0780 -3.6873  0.0003"                                                      
[21] "sar1      -0.1652 0.0908 -1.8203  0.0706"                                                      
[22] "sar2      -0.0364 0.0896 -0.4065  0.6849"                                                      
[23] "constant   0.0067 0.0080  0.8386  0.4029"                                                      
[24] ""                                                                                              
[25] "$AIC"                                                                                          
[26] "[1] -0.5086453"                                                                                
[27] ""                                                                                              
[28] "$AICc"                                                                                         
[29] "[1] -0.5055015"                                                                                
[30] ""                                                                                              
[31] "$BIC"                                                                                          
[32] "[1] -0.3779509"                                                                                
Series: ts 
ARIMA(1,1,1)(2,1,1)[4] 

Coefficients:
         ar1      ma1     sar1     sar2     sma1
      0.8182  -1.0000  -0.1960  -0.1040  -0.8524
s.e.  0.0520   0.0256   0.0993   0.0982   0.0615

sigma^2 = 0.03223:  log likelihood = 44.78
AIC=-77.55   AICc=-77.01   BIC=-58.99

Training set error measures:
                      ME      RMSE       MAE      MPE     MAPE      MASE
Training set -0.01547569 0.1740971 0.1211176 37.62815 140.1717 0.5557939
                     ACF1
Training set -0.009057403


Equation for SARIMA(1,1,1)(2,1,1)[4]:

\[(1 - \phi B)(1 - B)(1 - \Phi_1 B^s)(1 - \Phi_2 B^{2s})(1 - B^s)X_t = (1 + \theta B)(1 + \Theta_1 B^s)W_t\]


Forecasting


Benchmark Method

SARIMA Model Accuracy Metrics:
                      ME      RMSE       MAE      MPE     MAPE      MASE
Training set -0.01547569 0.1740971 0.1211176 37.62815 140.1717 0.5557939
                     ACF1
Training set -0.009057403

Mean Model Accuracy Metrics:
                        ME      RMSE       MAE      MPE     MAPE     MASE
Training set -1.183749e-17 0.3317336 0.2505159 3.214205 311.7334 1.149587
                  ACF1
Training set 0.8236012

Naive Model Accuracy Metrics:
                      ME     RMSE       MAE     MPE     MAPE     MASE
Training set 0.006370865 0.188809 0.1231996 17.7233 112.8604 0.565348
                    ACF1
Training set -0.02926909

Seasonal Naive Model Accuracy Metrics:
                    ME      RMSE       MAE      MPE     MAPE MASE      ACF1
Training set 0.0353719 0.3282686 0.2179181 59.57998 334.7433    1 0.6614068

Random Walk with Drift Model Accuracy Metrics:
                      ME      RMSE       MAE     MPE     MAPE      MASE
Training set 2.94903e-17 0.1887014 0.1234148 20.7863 112.9409 0.5663355
                    ACF1
Training set -0.02926909

Model with the best Accuracy Metrics:
 SARIMA SARIMA SARIMA Mean Naive SARIMA Drift


Cross Validation

One-Step Ahead Cross Validation
Model 1 - MAE: 0.09014297 MSE: 0.01947199 
Model 2 - MAE: 0.0967174 MSE: 0.02260441 
Model 1 performs better on both MAE and MSE.

Data Processing


Stationarity Check

Augmented Dickey-Fuller Test Results:
Test Statistic: -3.805526   P-value: 0.02041494
The time series is stationary based on the ADF test.


Model Fitting

Minimum AIC: 2, 1, 2, 2, 1, 1, -0.473054691655761, 24.2769469147983, 0.462010243409175 
Minimum BIC: 1, 1, 1, 2, 1, 1, 5.03140449969021, 23.5939057045308, 5.56986603815175 
Minimum AICc: 2, 1, 2, 2, 1, 1, -0.473054691655761, 24.2769469147983, 0.462010243409175 
Series: ts 
ARIMA(0,1,0)(2,0,0)[4] 

Coefficients:
         sar1     sar2
      -0.2138  -0.2045
s.e.   0.0909   0.0933

sigma^2 = 0.05475:  log likelihood = 6.38
AIC=-6.75   AICc=-6.61   BIC=2.6


Model Diagnostics

 [1] "initial  value -1.584642 "                                                                     
 [2] "iter   2 value -1.628709"                                                                      
 [3] "iter   3 value -1.628981"                                                                      
 [4] "iter   4 value -1.629047"                                                                      
 [5] "iter   5 value -1.629047"                                                                      
 [6] "iter   6 value -1.629047"                                                                      
 [7] "iter   6 value -1.629047"                                                                      
 [8] "iter   6 value -1.629047"                                                                      
 [9] "final  value -1.629047 "                                                                       
[10] "converged"                                                                                     
[11] "initial  value -1.454586 "                                                                     
[12] "iter   2 value -1.457024"                                                                      
[13] "iter   3 value -1.457204"                                                                      
[14] "iter   4 value -1.457212"                                                                      
[15] "iter   5 value -1.457212"                                                                      
[16] "iter   5 value -1.457212"                                                                      
[17] "iter   5 value -1.457212"                                                                      
[18] "final  value -1.457212 "                                                                       
[19] "converged"                                                                                     
[20] "$fit"                                                                                          
[21] ""                                                                                              
[22] "Call:"                                                                                         
[23] "arima(x = xdata, order = c(p, d, q), seasonal = list(order = c(P, D, Q), period = S), "        
[24] "    xreg = constant, transform.pars = trans, fixed = fixed, optim.control = list(trace = trc, "
[25] "        REPORT = 1, reltol = tol))"                                                            
[26] ""                                                                                              
[27] "Coefficients:"                                                                                 
[28] "         sar1     sar2  constant"                                                              
[29] "      -0.2142  -0.2046    0.0022"                                                              
[30] "s.e.   0.0910   0.0933    0.0128"                                                              
[31] ""                                                                                              
[32] "sigma^2 estimated as 0.05408:  log likelihood = 6.39,  aic = -4.78"                            
[33] ""                                                                                              
[34] "$degrees_of_freedom"                                                                           
[35] "[1] 164"                                                                                       
[36] ""                                                                                              
[37] "$ttable"                                                                                       
[38] "         Estimate     SE t.value p.value"                                                      
[39] "sar1      -0.2142 0.0910 -2.3553  0.0197"                                                      
[40] "sar2      -0.2046 0.0933 -2.1921  0.0298"                                                      
[41] "constant   0.0022 0.0128  0.1753  0.8611"                                                      
[42] ""                                                                                              
[43] "$AIC"                                                                                          
[44] "[1] -0.02864336"                                                                               
[45] ""                                                                                              
[46] "$AICc"                                                                                         
[47] "[1] -0.02776168"                                                                               
[48] ""                                                                                              
[49] "$BIC"                                                                                          
[50] "[1] 0.04603913"                                                                                
[51] ""                                                                                              

  [1] "initial  value -1.153352 "                                                                                 
  [2] "iter   2 value -1.432400"                                                                                  
  [3] "iter   3 value -1.542238"                                                                                  
  [4] "iter   4 value -1.556935"                                                                                  
  [5] "iter   5 value -1.578526"                                                                                  
  [6] "iter   6 value -1.583659"                                                                                  
  [7] "iter   7 value -1.586425"                                                                                  
  [8] "iter   8 value -1.588110"                                                                                  
  [9] "iter   9 value -1.590271"                                                                                  
 [10] "iter  10 value -1.592138"                                                                                  
 [11] "iter  11 value -1.594681"                                                                                  
 [12] "iter  12 value -1.596694"                                                                                  
 [13] "iter  13 value -1.601751"                                                                                  
 [14] "iter  14 value -1.604679"                                                                                  
 [15] "iter  15 value -1.610029"                                                                                  
 [16] "iter  16 value -1.614741"                                                                                  
 [17] "iter  17 value -1.614838"                                                                                  
 [18] "iter  18 value -1.615068"                                                                                  
 [19] "iter  19 value -1.615308"                                                                                  
 [20] "iter  20 value -1.616168"                                                                                  
 [21] "iter  21 value -1.618286"                                                                                  
 [22] "iter  22 value -1.619779"                                                                                  
 [23] "iter  23 value -1.621335"                                                                                  
 [24] "iter  24 value -1.622036"                                                                                  
 [25] "iter  25 value -1.623285"                                                                                  
 [26] "iter  26 value -1.623420"                                                                                  
 [27] "iter  27 value -1.623546"                                                                                  
 [28] "iter  28 value -1.623566"                                                                                  
 [29] "iter  29 value -1.623567"                                                                                  
 [30] "iter  30 value -1.623567"                                                                                  
 [31] "iter  31 value -1.623568"                                                                                  
 [32] "iter  31 value -1.623568"                                                                                  
 [33] "final  value -1.623568 "                                                                                   
 [34] "converged"                                                                                                 
 [35] "initial  value -1.411246 "                                                                                 
 [36] "iter   2 value -1.417760"                                                                                  
 [37] "iter   3 value -1.423349"                                                                                  
 [38] "iter   4 value -1.424629"                                                                                  
 [39] "iter   5 value -1.427692"                                                                                  
 [40] "iter   6 value -1.428452"                                                                                  
 [41] "iter   7 value -1.428780"                                                                                  
 [42] "iter   8 value -1.428960"                                                                                  
 [43] "iter   9 value -1.429343"                                                                                  
 [44] "iter  10 value -1.429644"                                                                                  
 [45] "iter  11 value -1.429796"                                                                                  
 [46] "iter  12 value -1.429901"                                                                                  
 [47] "iter  13 value -1.429961"                                                                                  
 [48] "iter  14 value -1.430077"                                                                                  
 [49] "iter  15 value -1.430343"                                                                                  
 [50] "iter  16 value -1.430785"                                                                                  
 [51] "iter  17 value -1.431139"                                                                                  
 [52] "iter  18 value -1.431454"                                                                                  
 [53] "iter  19 value -1.432390"                                                                                  
 [54] "iter  20 value -1.433279"                                                                                  
 [55] "iter  21 value -1.434767"                                                                                  
 [56] "iter  22 value -1.435427"                                                                                  
 [57] "iter  23 value -1.438964"                                                                                  
 [58] "iter  24 value -1.445525"                                                                                  
 [59] "iter  25 value -1.445894"                                                                                  
 [60] "iter  26 value -1.446931"                                                                                  
 [61] "iter  27 value -1.447249"                                                                                  
 [62] "iter  28 value -1.447332"                                                                                  
 [63] "iter  29 value -1.447346"                                                                                  
 [64] "iter  30 value -1.447347"                                                                                  
 [65] "iter  31 value -1.447347"                                                                                  
 [66] "iter  32 value -1.447350"                                                                                  
 [67] "iter  33 value -1.447354"                                                                                  
 [68] "iter  34 value -1.447372"                                                                                  
 [69] "iter  35 value -1.447480"                                                                                  
 [70] "iter  36 value -1.447707"                                                                                  
 [71] "iter  37 value -1.447878"                                                                                  
 [72] "iter  38 value -1.448033"                                                                                  
 [73] "iter  39 value -1.448932"                                                                                  
 [74] "iter  40 value -1.449332"                                                                                  
 [75] "iter  41 value -1.449415"                                                                                  
 [76] "iter  42 value -1.449518"                                                                                  
 [77] "iter  43 value -1.449574"                                                                                  
 [78] "iter  44 value -1.449704"                                                                                  
 [79] "iter  45 value -1.449988"                                                                                  
 [80] "iter  46 value -1.450732"                                                                                  
 [81] "iter  47 value -1.451444"                                                                                  
 [82] "iter  48 value -1.453977"                                                                                  
 [83] "iter  49 value -1.459680"                                                                                  
 [84] "iter  50 value -1.460383"                                                                                  
 [85] "iter  51 value -1.462770"                                                                                  
 [86] "iter  52 value -1.463032"                                                                                  
 [87] "iter  53 value -1.463517"                                                                                  
 [88] "iter  54 value -1.463604"                                                                                  
 [89] "iter  55 value -1.463778"                                                                                  
 [90] "iter  56 value -1.464020"                                                                                  
 [91] "iter  57 value -1.464266"                                                                                  
 [92] "iter  58 value -1.464640"                                                                                  
 [93] "iter  59 value -1.465612"                                                                                  
 [94] "iter  60 value -1.466519"                                                                                  
 [95] "iter  61 value -1.467018"                                                                                  
 [96] "iter  62 value -1.469013"                                                                                  
 [97] "iter  63 value -1.469387"                                                                                  
 [98] "iter  64 value -1.469429"                                                                                  
 [99] "iter  65 value -1.469440"                                                                                  
[100] "iter  66 value -1.469459"                                                                                  
[101] "iter  67 value -1.469461"                                                                                  
[102] "iter  68 value -1.469465"                                                                                  
[103] "iter  69 value -1.469466"                                                                                  
[104] "iter  70 value -1.469467"                                                                                  
[105] "iter  71 value -1.469467"                                                                                  
[106] "iter  72 value -1.469467"                                                                                  
[107] "iter  73 value -1.469467"                                                                                  
[108] "iter  74 value -1.469467"                                                                                  
[109] "iter  75 value -1.469467"                                                                                  
[110] "iter  76 value -1.469468"                                                                                  
[111] "iter  77 value -1.469468"                                                                                  
[112] "iter  78 value -1.469469"                                                                                  
[113] "iter  79 value -1.469469"                                                                                  
[114] "iter  80 value -1.469469"                                                                                  
[115] "iter  80 value -1.469469"                                                                                  
[116] "iter  80 value -1.469469"                                                                                  
[117] "final  value -1.469469 "                                                                                   
[118] "converged"                                                                                                 
[119] "$fit"                                                                                                      
[120] ""                                                                                                          
[121] "Call:"                                                                                                     
[122] "arima(x = xdata, order = c(p, d, q), seasonal = list(order = c(P, D, Q), period = S), "                    
[123] "    include.mean = !no.constant, transform.pars = trans, fixed = fixed, optim.control = list(trace = trc, "
[124] "        REPORT = 1, reltol = tol))"                                                                        
[125] ""                                                                                                          
[126] "Coefficients:"                                                                                             
[127] "          ar1     ar2      ma1      ma2     sar1     sar2     sma1"                                        
[128] "      -0.1820  0.8030  -0.1146  -0.8854  -0.2283  -0.2178  -0.9541"                                        
[129] "s.e.   0.0611  0.0563   0.0685   0.0667   0.0991   0.0979   0.0638"                                        
[130] ""                                                                                                          
[131] "sigma^2 estimated as 0.04746:  log likelihood = 8.24,  aic = -0.47"                                        
[132] ""                                                                                                          
[133] "$degrees_of_freedom"                                                                                       
[134] "[1] 156"                                                                                                   
[135] ""                                                                                                          
[136] "$ttable"                                                                                                   
[137] "     Estimate     SE  t.value p.value"                                                                     
[138] "ar1   -0.1820 0.0611  -2.9789  0.0034"                                                                     
[139] "ar2    0.8030 0.0563  14.2623  0.0000"                                                                     
[140] "ma1   -0.1146 0.0685  -1.6739  0.0961"                                                                     
[141] "ma2   -0.8854 0.0667 -13.2651  0.0000"                                                                     
[142] "sar1  -0.2283 0.0991  -2.3029  0.0226"                                                                     
[143] "sar2  -0.2178 0.0979  -2.2244  0.0276"                                                                     
[144] "sma1  -0.9541 0.0638 -14.9487  0.0000"                                                                     
[145] ""                                                                                                          
[146] "$AIC"                                                                                                      
[147] "[1] -0.002902176"                                                                                          
[148] ""                                                                                                          
[149] "$AICc"                                                                                                     
[150] "[1] 0.001530834"                                                                                           
[151] ""                                                                                                          
[152] "$BIC"                                                                                                      
[153] "[1] 0.1489383"                                                                                             
[154] ""                                                                                                          

  [1] "initial  value -1.156609 "                                                                                 
  [2] "iter   2 value -1.448236"                                                                                  
  [3] "iter   3 value -1.503849"                                                                                  
  [4] "iter   4 value -1.521930"                                                                                  
  [5] "iter   5 value -1.561078"                                                                                  
  [6] "iter   6 value -1.589149"                                                                                  
  [7] "iter   7 value -1.596845"                                                                                  
  [8] "iter   8 value -1.597994"                                                                                  
  [9] "iter   9 value -1.598385"                                                                                  
 [10] "iter  10 value -1.598396"                                                                                  
 [11] "iter  11 value -1.598402"                                                                                  
 [12] "iter  12 value -1.598495"                                                                                  
 [13] "iter  13 value -1.598515"                                                                                  
 [14] "iter  14 value -1.598521"                                                                                  
 [15] "iter  15 value -1.598526"                                                                                  
 [16] "iter  16 value -1.598527"                                                                                  
 [17] "iter  17 value -1.598532"                                                                                  
 [18] "iter  18 value -1.598544"                                                                                  
 [19] "iter  19 value -1.598552"                                                                                  
 [20] "iter  20 value -1.598555"                                                                                  
 [21] "iter  21 value -1.598555"                                                                                  
 [22] "iter  21 value -1.598555"                                                                                  
 [23] "iter  21 value -1.598555"                                                                                  
 [24] "final  value -1.598555 "                                                                                   
 [25] "converged"                                                                                                 
 [26] "initial  value -1.373356 "                                                                                 
 [27] "iter   2 value -1.380275"                                                                                  
 [28] "iter   3 value -1.418737"                                                                                  
 [29] "iter   4 value -1.420191"                                                                                  
 [30] "iter   5 value -1.422171"                                                                                  
 [31] "iter   6 value -1.422187"                                                                                  
 [32] "iter   7 value -1.422236"                                                                                  
 [33] "iter   8 value -1.422535"                                                                                  
 [34] "iter   9 value -1.422801"                                                                                  
 [35] "iter  10 value -1.423264"                                                                                  
 [36] "iter  11 value -1.423836"                                                                                  
 [37] "iter  12 value -1.425616"                                                                                  
 [38] "iter  13 value -1.427219"                                                                                  
 [39] "iter  14 value -1.428113"                                                                                  
 [40] "iter  15 value -1.428671"                                                                                  
 [41] "iter  16 value -1.428859"                                                                                  
 [42] "iter  17 value -1.436250"                                                                                  
 [43] "iter  18 value -1.437695"                                                                                  
 [44] "iter  19 value -1.438645"                                                                                  
 [45] "iter  20 value -1.438726"                                                                                  
 [46] "iter  21 value -1.438753"                                                                                  
 [47] "iter  22 value -1.438826"                                                                                  
 [48] "iter  23 value -1.438963"                                                                                  
 [49] "iter  24 value -1.439319"                                                                                  
 [50] "iter  25 value -1.439923"                                                                                  
 [51] "iter  26 value -1.440203"                                                                                  
 [52] "iter  27 value -1.440304"                                                                                  
 [53] "iter  28 value -1.440312"                                                                                  
 [54] "iter  29 value -1.440312"                                                                                  
 [55] "iter  30 value -1.440312"                                                                                  
 [56] "iter  31 value -1.440312"                                                                                  
 [57] "iter  32 value -1.440312"                                                                                  
 [58] "iter  33 value -1.440313"                                                                                  
 [59] "iter  34 value -1.440313"                                                                                  
 [60] "iter  35 value -1.440314"                                                                                  
 [61] "iter  36 value -1.440314"                                                                                  
 [62] "iter  37 value -1.440315"                                                                                  
 [63] "iter  38 value -1.440315"                                                                                  
 [64] "iter  38 value -1.440315"                                                                                  
 [65] "iter  38 value -1.440315"                                                                                  
 [66] "final  value -1.440315 "                                                                                   
 [67] "converged"                                                                                                 
 [68] "$fit"                                                                                                      
 [69] ""                                                                                                          
 [70] "Call:"                                                                                                     
 [71] "arima(x = xdata, order = c(p, d, q), seasonal = list(order = c(P, D, Q), period = S), "                    
 [72] "    include.mean = !no.constant, transform.pars = trans, fixed = fixed, optim.control = list(trace = trc, "
 [73] "        REPORT = 1, reltol = tol))"                                                                        
 [74] ""                                                                                                          
 [75] "Coefficients:"                                                                                             
 [76] "          ar1     ma1     sar1     sar2     sma1"                                                          
 [77] "      -0.9902  0.8509  -0.2971  -0.2563  -0.9561"                                                          
 [78] "s.e.   0.0226  0.0712   0.0916   0.0946   0.0585"                                                          
 [79] ""                                                                                                          
 [80] "sigma^2 estimated as 0.05234:  log likelihood = 3.48,  aic = 5.03"                                         
 [81] ""                                                                                                          
 [82] "$degrees_of_freedom"                                                                                       
 [83] "[1] 158"                                                                                                   
 [84] ""                                                                                                          
 [85] "$ttable"                                                                                                   
 [86] "     Estimate     SE  t.value p.value"                                                                     
 [87] "ar1   -0.9902 0.0226 -43.8806  0.0000"                                                                     
 [88] "ma1    0.8509 0.0712  11.9567  0.0000"                                                                     
 [89] "sar1  -0.2971 0.0916  -3.2442  0.0014"                                                                     
 [90] "sar2  -0.2563 0.0946  -2.7084  0.0075"                                                                     
 [91] "sma1  -0.9561 0.0585 -16.3309  0.0000"                                                                     
 [92] ""                                                                                                          
 [93] "$AIC"                                                                                                      
 [94] "[1] 0.03086751"                                                                                            
 [95] ""                                                                                                          
 [96] "$AICc"                                                                                                     
 [97] "[1] 0.03321209"                                                                                            
 [98] ""                                                                                                          
 [99] "$BIC"                                                                                                      
[100] "[1] 0.1447479"                                                                                             
[101] ""                                                                                                          
Series: ts 
ARIMA(2,1,2)(2,1,1)[4] 

Coefficients:
          ar1     ar2      ma1      ma2     sar1     sar2     sma1
      -0.1820  0.8030  -0.1146  -0.8854  -0.2283  -0.2178  -0.9541
s.e.   0.0611  0.0563   0.0685   0.0667   0.0991   0.0979   0.0638

sigma^2 = 0.04959:  log likelihood = 8.24
AIC=-0.47   AICc=0.46   BIC=24.28

Training set error measures:
                      ME      RMSE       MAE      MPE     MAPE      MASE
Training set 0.004070017 0.2145945 0.1528131 104.5816 243.6773 0.5254351
                  ACF1
Training set 0.0078855


Equation for SARIMA(2,1,2)(2,1,1)[4]:

\[(1 - \phi_1 B - \phi_2 B^2)(1 - B)(1 - \Phi_1 B^s - \Phi_2 B^{2s})(1 - B^s)X_t = (1 + \theta_1 B + \theta_2 B^2)(1 + \Theta_1 B^s)W_t\]


Forecasting


Benchmark Method

SARIMA Model Accuracy Metrics:
                      ME      RMSE       MAE      MPE     MAPE      MASE
Training set 0.004070017 0.2145945 0.1528131 104.5816 243.6773 0.5254351
                  ACF1
Training set 0.0078855

Mean Model Accuracy Metrics:
                       ME      RMSE       MAE      MPE     MAPE      MASE
Training set 8.108929e-19 0.3310926 0.2327435 161.2462 176.9307 0.8002689
                  ACF1
Training set 0.7392247

Naive Model Accuracy Metrics:
                      ME      RMSE      MAE      MPE     MAPE      MASE
Training set 0.002112496 0.2387966 0.158006 47.22492 225.3856 0.5432904
                   ACF1
Training set -0.1433554

Seasonal Naive Model Accuracy Metrics:
                     ME      RMSE       MAE      MPE     MAPE MASE      ACF1
Training set 0.01693984 0.4100165 0.2908316 240.2875 374.3727    1 0.6545677

Random Walk with Drift Model Accuracy Metrics:
                        ME      RMSE       MAE      MPE     MAPE      MASE
Training set -3.941913e-17 0.2387872 0.1581026 42.26205 229.5025 0.5436225
                   ACF1
Training set -0.1433554

Model with the best Accuracy Metrics:
 Drift SARIMA SARIMA Drift Mean SARIMA Drift


Cross Validation

One-Step Ahead Cross Validation
Model 1 - MAE: 0.132274 MSE: 0.03496468 
Model 2 - MAE: 0.1428235 MSE: 0.03843426 
Model 1 performs better on both MAE and MSE.